Short version: does anyone know of any good sources on class-forcing over E-closed, non-admissible sets?

Longer version: A problem I'm working on has reached an interesting conclusion - I've managed to show that, under Appropriate Hypotheses, there is a countable ordinal $\gamma$ which is the height of a transitive set with some surprisingly nice determinacy properties. The ordinal in question, though, is annoyingly high, and so I'm trying to bring it down a bit. The problem is that building the transitive set in question involves a somewhat messy iterated forcing over $L_\gamma$. If all the pieces of the iteration are sets in $L_\gamma$, then everything is nice and easy - but ensuring that of course is what makes $\gamma$ so big. I think I've figured out how to bump $\gamma$ down a fair ways via class forcing over admissible sets; however, the real accomplishment would be to bump $\gamma$ down even further and show that the forcing behaves nicely even when $L_\gamma$ is non-admissible but E-closed!

. . . Which, unfortunately, is a bit of a non-starter, since I can find absolutely nothing about class forcing over E-closed, non-admissible sets.

To give some context for why I don't want to attack this entirely from scratch, let me briefly describe the situation. Set forcing, of course, preserves ZFC and ZF - which raises the question of what fragments T $\subset $ ZFC are also preserved by set forcing. With some care, but not too much difficulty, we can show that set forcing preserves the very tiny fragment KP - that is, a set forcing extension of an admissible set is again admissible. (I believe the first person to observe this was Ershov.) Class forcing is more complicated, but not too terrible.

At the E-closed but not admissible level, however, everything goes bonkers. Let $E(\omega_1)$ denote the least E-closed set containing $\omega_1$. Then there is a set forcing in $E(\omega_1)$ which does not preserve E-closedness!

(That forcing is the usual collapse of $\omega_1$ to $\omega$. The reason this does not preserve E-closedness is roughly: if $b$ is the real coding the generic collapse $G$ of $\omega_1$, then $E(\omega_1)[G]=E(b)$ if the former is E-closed. But $E(b)=L_{\omega_1^b}(b)$ is admissible since $b\subset\omega$; so $E(\omega_1)$, which has the same height as $E(b)$, must also be admissible. However, $E(\omega_1)$ is not admissible, since it admits divergence witnesses.)

Now, there are sources on class forcing over admissible sets; Jensen has some notes about this, and Friedman's article in an old "Philosophy, Logic, and Methodology of Science" volume gives a good description of almost disjoint coding in this context. Similarly, there are sources on forcing over E-closed sets - besides an article by Sacks and an article by Sacks and Slaman, there is the thesis of Sherry Marcus. However, as far as I can tell, nobody treats the problem of class forcing in an E-closed, non-admissible context.

So the question stands: does anyone know a source on class forcing over E-closed but non-admissible sets?